Acosta was born in Havana, Cuba, on 2 June 1973, the eleventh and last child in an impoverished Havana family. Pedro Acosta, his father was a truck driver, and Dulce Maria Quesada, his mother, often suffered from health problems. Acosta grew up with no toys, sometimes went shoeless, and did not even have a birthday cake until he turned 23. The streets of his neighborhood provided plenty of entertainment, however, and he spent his time playing football, break-dancing, and raiding nearby mango groves with his friends. He was an over-energetic child, and his father felt that his youngest son would soon land in serious trouble. Dance training at one of the state-funded schools, his father decided, would teach the boy discipline and provide him with a free lunch every day. He studied ballet at the Cuban National Ballet School with many influential teachers including Ramona de Sáa. In June 1991 he received his diploma with maximum qualifications and a gold medal.

Acosta, of mixed Spanish and African heritage, came to prominence in the early 1990s while still in his teens, and North American and European dance companies began offering him lead romantic roles over the next decade. After five years in Houston, Acosta joined London's Royal Ballet in 1998. With his fabled grace and athleticism, he has earned comparisons to Mikhail Baryshnikov or Rudolf Nureyev. A writer for London's Independent newspaper described Acosta as "a dancer who slashes across space faster than anyone else, who lacerates the air with shapes so clear and sharp they seem to throw off sparks".[2]

In November 1993 he was invited by Ben Stevenson, the artistic director of Houston Ballet, to join the company as a principal dancer, and he made his American stage debut as the Prince in The Nutcracker. Following this, his repertory included:

In recent years, Acosta has had a career as an international guest artist, appearing in the US, Russia, the Netherlands, Chile, Argentina, Greece, Japan, Italy, Germany and France. In June 2008 he guested for the first time with the Australian Ballet, in Jerome Robbins' Afternoon of a Faun, partnered by AB principal Kirsty Martin.

In 1997 CBS News filmed Acosta for a special report for their programme 60 Minutes (broadcast in January 1998), following him during his performances for the Houston Ballet and relaxing with friends and family in Cuba.

Television performances include two live BBC broadcasts from The Royal Opera House: the Opening Celebration in December 1999 when Acosta performed the Man's Solo in Le Corsaire, and in February 2000 the role of Franz in Ninette de Valois' Coppélia. Most recently Acosta was featured in "The Reluctant Ballet Dancer", a programme in the Imagine series shown on BBC1 on 9 July 2003 and presented by Alan Yentob. He was also interviewed for the BBC's HardTalk programme which was broadcast on Christmas Day 2003. In 2006 he was filmed in the BBC studios performing a dance from Tocororo, which was broadcast on the Newsnight programme.

Tocororo is Acosta's own ballet and has been touring theatres worldwide to high acclaim.[citation needed]

Tocororo is the story of a young Cuban boy who leaves his family and home in the Cuban countryside to find a new life in a city. It is loosely based on Acosta's own life experiences. It is set to original music by Miguel Nuñez, which blends popular and symphonic Cuban styles and is performed live on stage by a group of five Cuban musicians.

Acosta chose the 17-strong company from dancers in Danza Contemporanea de Cuba, the Cuban National Ballet and Conjunto Folklorico Nacional de Cuba. The "young protagonist" is played by Yonah Acosta, Acosta's younger nephew, and for six performances the central role was played by José Oduardo Perez.

No Way Home - a Cuban Dancer's Story by Carlos Acosta was published in 2007 by HarperCollins UK, Scribner US and Schott in Germany. He tells the story of his rise from a poor boy in Havana to becoming the leading male ballet dancer in the World, chosen by the Bolshoi Ballet to dance the lead in Spartacas that year.